Source code for botorch.models.contextual_multioutput
from gpytorch.constraints import Interval
from gpytorch.distributions.multivariate_normal import MultivariateNormal
from gpytorch.kernels.rbf_kernel import RBFKernel
-from gpytorch.likelihoods.gaussian_likelihood import FixedNoiseGaussianLikelihood
from linear_operator.operators import InterpolatedLinearOperator, LinearOperator
from torch import Tensor
from torch.nn import ModuleList
[docs]class LCEMGP(MultiTaskGP):
- r"""The Multi-Task GP with the latent context embedding multioutput
- (LCE-M) kernel.
+ r"""The Multi-Task GP with the latent context embedding multioutput (LCE-M)
+ kernel. See [Feng2020HDCPS]_ for a reference on the model and its use in Bayesian
+ optimization.
+
"""
def __init__(
@@ -52,6 +63,7 @@
Source code for botorch.models.contextual_multioutput
train_X: Tensor,
train_Y: Tensor,
task_feature: int,
+ train_Yvar: Optional[Tensor] = None,
context_cat_feature: Optional[Tensor] = None,
context_emb_feature: Optional[Tensor] = None,
embs_dim_list: Optional[List[int]] = None,
@@ -64,6 +76,9 @@
Source code for botorch.models.contextual_multioutput
train_X: (n x d) X training data.
train_Y: (n x 1) Y training data.
task_feature: Column index of train_X to get context indices.
+ train_Yvar: An optional (n x 1) tensor of observed variances of each
+ training Y. If None, we infer the noise. Note that the inferred noise
+ is common across all tasks.
context_cat_feature: (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
number of categorical variables. If None, task indices will
@@ -75,11 +90,13 @@
Source code for botorch.models.contextual_multioutput
for each categorical variable.
output_tasks: A list of task indices for which to compute model
outputs for. If omitted, return outputs for all task indices.
+
"""
super().__init__(
train_X=train_X,
train_Y=train_Y,
task_feature=task_feature,
+ train_Yvar=train_Yvar,
output_tasks=output_tasks,
input_transform=input_transform,
outcome_transform=outcome_transform,
@@ -149,6 +166,7 @@
diff --git a/v/latest/api/_modules/botorch/models/contextual_multioutput/index.html b/v/latest/api/_modules/botorch/models/contextual_multioutput/index.html
index 16a76051ee..7bce3e456f 100644
--- a/v/latest/api/_modules/botorch/models/contextual_multioutput/index.html
+++ b/v/latest/api/_modules/botorch/models/contextual_multioutput/index.html
@@ -27,6 +27,16 @@ Source code for botorch.models.contextual_multioutput
Args:
task_idcs: (n x 1) or (b x n x 1) task indices tensor
+
"""
covar_matrix = self._eval_context_covar()
return InterpolatedLinearOperator(
@@ -173,6 +191,8 @@
Source code for botorch.models.contextual_multioutput
[docs]class FixedNoiseLCEMGP(LCEMGP):
r"""The Multi-Task GP the latent context embedding multioutput
(LCE-M) kernel, with known observation noise.
+
+ DEPRECATED: Please use `LCEMGP` with `train_Yvar` instead.
"""
def __init__(
@@ -190,7 +210,7 @@
Source code for botorch.models.contextual_multioutput
Args:
train_X: (n x d) X training data.
train_Y: (n x 1) Y training data.
- train_Yvar: (n x 1) Noise variances of each training Y.
+ train_Yvar: (n x 1) Observed variances of each training Y.
task_feature: Column index of train_X to get context indices.
context_cat_feature: (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
@@ -203,19 +223,26 @@
Source code for botorch.models.contextual_multioutput
1 for each categorical variable.
output_tasks: A list of task indices for which to compute model
outputs for. If omitted, return outputs for all task indices.
+
"""
- self._validate_tensor_args(X=train_X, Y=train_Y, Yvar=train_Yvar)
+ warnings.warn(
+ "`FixedNoiseLCEMGP` has been deprecated and will be removed in a "
+ "future release. Please use the `LCEMGP` model instead. "
+ "When `train_Yvar` is specified, `LCEMGP` behaves the same "
+ "as the `FixedNoiseLCEMGP`.",
+ DeprecationWarning,
+ )
+
super().__init__(
train_X=train_X,
train_Y=train_Y,
task_feature=task_feature,
+ train_Yvar=train_Yvar,
context_cat_feature=context_cat_feature,
context_emb_feature=context_emb_feature,
embs_dim_list=embs_dim_list,
output_tasks=output_tasks,
- )
- self.likelihood = FixedNoiseGaussianLikelihood(noise=train_Yvar)
- self.to(train_X)
+ )Source code for botorch.models.contextual_multioutput
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
+r"""
+References
+
+.. [Feng2020HDCPS]
+ Q. Feng, B. Latham, H. Mao and E. Backshy. High-Dimensional Contextual Policy
+ Search with Unknown Context Rewards using Bayesian Optimization.
+ Advances in Neural Information Processing Systems 33, NeurIPS 2020.
+"""
+
+import warnings
from typing import List, Optional
import torch
@@ -36,15 +46,16 @@
Source code for botorch.models.contextual_multioutput
from gpytorch.constraints import Interval
from gpytorch.distributions.multivariate_normal import MultivariateNormal
from gpytorch.kernels.rbf_kernel import RBFKernel
-from gpytorch.likelihoods.gaussian_likelihood import FixedNoiseGaussianLikelihood
from linear_operator.operators import InterpolatedLinearOperator, LinearOperator
from torch import Tensor
from torch.nn import ModuleList
[docs]class LCEMGP(MultiTaskGP):
- r"""The Multi-Task GP with the latent context embedding multioutput
- (LCE-M) kernel.
+ r"""The Multi-Task GP with the latent context embedding multioutput (LCE-M)
+ kernel. See [Feng2020HDCPS]_ for a reference on the model and its use in Bayesian
+ optimization.
+
"""
def __init__(
@@ -52,6 +63,7 @@
Source code for botorch.models.contextual_multioutput
train_X: Tensor,
train_Y: Tensor,
task_feature: int,
+ train_Yvar: Optional[Tensor] = None,
context_cat_feature: Optional[Tensor] = None,
context_emb_feature: Optional[Tensor] = None,
embs_dim_list: Optional[List[int]] = None,
@@ -64,6 +76,9 @@
Source code for botorch.models.contextual_multioutput
train_X: (n x d) X training data.
train_Y: (n x 1) Y training data.
task_feature: Column index of train_X to get context indices.
+ train_Yvar: An optional (n x 1) tensor of observed variances of each
+ training Y. If None, we infer the noise. Note that the inferred noise
+ is common across all tasks.
context_cat_feature: (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
number of categorical variables. If None, task indices will
@@ -75,11 +90,13 @@ Models
Source code for botorch.models.contextual_multioutput
for each categorical variable.
output_tasks: A list of task indices for which to compute model
outputs for. If omitted, return outputs for all task indices.
+
"""
super().__init__(
train_X=train_X,
train_Y=train_Y,
task_feature=task_feature,
+ train_Yvar=train_Yvar,
output_tasks=output_tasks,
input_transform=input_transform,
outcome_transform=outcome_transform,
@@ -149,6 +166,7 @@
diff --git a/v/latest/api/models.html b/v/latest/api/models.html
index e0582fedbd..f598b4e6ae 100644
--- a/v/latest/api/models.html
+++ b/v/latest/api/models.html
@@ -2347,18 +2347,31 @@ Source code for botorch.models.contextual_multioutput
Args:
task_idcs: (n x 1) or (b x n x 1) task indices tensor
+
"""
covar_matrix = self._eval_context_covar()
return InterpolatedLinearOperator(
@@ -173,6 +191,8 @@
Source code for botorch.models.contextual_multioutput
[docs]class FixedNoiseLCEMGP(LCEMGP):
r"""The Multi-Task GP the latent context embedding multioutput
(LCE-M) kernel, with known observation noise.
+
+ DEPRECATED: Please use `LCEMGP` with `train_Yvar` instead.
"""
def __init__(
@@ -190,7 +210,7 @@
Source code for botorch.models.contextual_multioutput
Args:
train_X: (n x d) X training data.
train_Y: (n x 1) Y training data.
- train_Yvar: (n x 1) Noise variances of each training Y.
+ train_Yvar: (n x 1) Observed variances of each training Y.
task_feature: Column index of train_X to get context indices.
context_cat_feature: (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
@@ -203,19 +223,26 @@
Source code for botorch.models.contextual_multioutput
1 for each categorical variable.
output_tasks: A list of task indices for which to compute model
outputs for. If omitted, return outputs for all task indices.
+
"""
- self._validate_tensor_args(X=train_X, Y=train_Y, Yvar=train_Yvar)
+ warnings.warn(
+ "`FixedNoiseLCEMGP` has been deprecated and will be removed in a "
+ "future release. Please use the `LCEMGP` model instead. "
+ "When `train_Yvar` is specified, `LCEMGP` behaves the same "
+ "as the `FixedNoiseLCEMGP`.",
+ DeprecationWarning,
+ )
+
super().__init__(
train_X=train_X,
train_Y=train_Y,
task_feature=task_feature,
+ train_Yvar=train_Yvar,
context_cat_feature=context_cat_feature,
context_emb_feature=context_emb_feature,
embs_dim_list=embs_dim_list,
output_tasks=output_tasks,
- )
- self.likelihood = FixedNoiseGaussianLikelihood(noise=train_Yvar)
- self.to(train_X)
+ )Models
Contextual GP Models with Context Rewards¶
+References
+
+
+[Feng2020HDCPS]
+Q. Feng, B. Latham, H. Mao and E. Backshy. High-Dimensional Contextual Policy
+Search with Unknown Context Rewards using Bayesian Optimization.
+Advances in Neural Information Processing Systems 33, NeurIPS 2020.
+
+
-
-class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
+class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, train_Yvar=None, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
Bases: MultiTaskGP
-The Multi-Task GP with the latent context embedding multioutput
-(LCE-M) kernel.
+The Multi-Task GP with the latent context embedding multioutput (LCE-M)
+kernel. See [Feng2020HDCPS] for a reference on the model and its use in Bayesian
+optimization.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
task_feature (int) – Column index of train_X to get context indices.
train_Yvar (Tensor | None) – An optional (n x 1) tensor of observed variances of each
+training Y. If None, we infer the noise. Note that the inferred noise
+is common across all tasks.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
number of categorical variables. If None, task indices will
@@ -2424,12 +2437,13 @@
ModelsBases: LCEMGP
The Multi-Task GP the latent context embedding multioutput
(LCE-M) kernel, with known observation noise.
+DEPRECATED: Please use LCEMGP with train_Yvar instead.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
train_Yvar (Tensor) – (n x 1) Noise variances of each training Y.
train_Yvar (Tensor) – (n x 1) Observed variances of each training Y.
task_feature (int) – Column index of train_X to get context indices.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
@@ -2461,21 +2475,21 @@
Models
[burt2020svgp]
-(1,2,3,4)
+(1,2,3,4)
David R. Burt and Carl Edward Rasmussen and Mark van der Wilk,
Convergence of Sparse Variational Inference in Gaussian Process Regression,
Journal of Machine Learning Research, 2020,
http://jmlr.org/papers/v21/19-1015.html.
-[hensman2013svgp]
+[hensman2013svgp]
James Hensman and Nicolo Fusi and Neil D. Lawrence, Gaussian Processes
for Big Data, Proceedings of the 29th Conference on Uncertainty in
Artificial Intelligence, 2013, https://arxiv.org/abs/1309.6835.
[moss2023ipa]
-(1,2,3,4)
+(1,2,3,4)
Henry B. Moss and Sebastian W. Ober and Victor Picheny,
Inducing Point Allocation for Sparse Gaussian Processes
in High-Throughput Bayesian Optimization,Proceedings of
@@ -2595,12 +2609,12 @@
Models
class botorch.models.approximate_gp.SingleTaskVariationalGP(train_X, train_Y=None, likelihood=None, num_outputs=1, learn_inducing_points=True, covar_module=None, mean_module=None, variational_distribution=None, variational_strategy=<class 'gpytorch.variational.variational_strategy.VariationalStrategy'>, inducing_points=None, outcome_transform=None, input_transform=None, inducing_point_allocator=None)[source]¶
Bases: ApproximateGPyTorchModel
-A single-task variational GP model following [hensman2013svgp].
+A single-task variational GP model following [hensman2013svgp].
By default, the inducing points are initialized though the
-GreedyVarianceReduction of [burt2020svgp], which is known to be
+GreedyVarianceReduction of [burt2020svgp], which is known to be
effective for building globally accurate models. However, custom
inducing point allocators designed for specific down-stream tasks can also be
-provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction
+provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction
when the goal is to build a model suitable for standard BO.
A single-task variational GP using relatively strong priors on the Kernel
hyperparameters, which work best when covariates are normalized to the unit
@@ -2712,7 +2726,7 @@
Models
[Eriksson2021saasbo]
-(1,2,3)
+(1,2,3)
D. Eriksson, M. Jankowiak. High-Dimensional Bayesian Optimization
with Sparse Axis-Aligned Subspaces. Proceedings of the Thirty-
Seventh Conference on Uncertainty in Artificial Intelligence, 2021.
@@ -2773,7 +2787,7 @@ ModelsImplementation of the sparse axis-aligned subspace priors (SAAS) model.
The SAAS model uses sparsity-inducing priors to identify the most important
parameters. This model is suitable for high-dimensional BO with potentially
-hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
+hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
SaasPyroModel is not a standard BoTorch model; instead, it is used as
an input to SaasFullyBayesianSingleTaskGP. It is used as a default keyword
argument, and end users are not likely to need to instantiate or modify a
@@ -2902,7 +2916,7 @@
ModelsThis model assumes that the inputs have been normalized to [0, 1]^d and that
the output has been standardized to have zero mean and unit variance. You can
either normalize and standardize the data before constructing the model or use
-an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo]
+an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo]
with a Matern-5/2 kernel is used by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it
isn’t compatible with fit_gpytorch_model.
@@ -3127,7 +3141,7 @@ ModelsA fully Bayesian multi-task GP model with the SAAS prior.
This model assumes that the inputs have been normalized to [0, 1]^d and that the
output has been stratified standardized to have zero mean and unit variance for
-each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data
+each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data
kernel by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it
isn’t compatible with fit_gpytorch_model.
@@ -3581,14 +3595,14 @@ Model Components
class botorch.models.kernels.orthogonal_additive_kernel.OrthogonalAdditiveKernel(base_kernel, dim, quad_deg=32, second_order=False, batch_shape=None, dtype=None, device=None, coeff_constraint=Positive())[source]¶
Bases: Kernel
-Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though
+
Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though
only for the case of Gaussian base kernels with a Gaussian input data distribution.
The implementation here generalizes OAKs to arbitrary base kernels by using a
Gauss-Legendre quadrature approximation to the required one-dimensional integrals
involving the base kernels.
-[Lu2022additive]
+[Lu2022additive]
X. Lu, A. Boukouvalas, and J. Hensman. Additive Gaussian processes revisited.
Proceedings of the 39th International Conference on Machine Learning. Jul 2022.
@@ -3818,7 +3832,7 @@ Transforms
-[eriksson2021scalable]
+[eriksson2021scalable]
D. Eriksson, M. Poloczek. Scalable Constrained Bayesian Optimization.
International Conference on Artificial Intelligence and Statistics. PMLR, 2021,
http://proceedings.mlr.press/v130/eriksson21a.html
@@ -4257,7 +4271,7 @@ Transformsclass botorch.models.transforms.outcome.Bilog(outputs=None)[source]¶
Bases: OutcomeTransform
Bilog-transform outcomes.
-The Bilog transform [eriksson2021scalable] is useful for modeling outcome
+
The Bilog transform [eriksson2021scalable] is useful for modeling outcome
constraints as it magnifies values near zero and flattens extreme values.
Bilog-transform outcomes.
@@ -4602,7 +4616,7 @@ Transforms[Daulton2022bopr]. This transformation supports differentiable
+proposed in [Daulton2022bopr]. This transformation supports differentiable
approximate rounding (currently only for integers). The rounding function
is approximated with a piece-wise function where each piece is a hyperbolic
tangent function.
@@ -4774,8 +4788,8 @@ Transforms[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition
-function introduced in [Cakmak2020risk] can be found at
+[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition
+function introduced in [Cakmak2020risk] can be found at
https://botorch.org/tutorials/risk_averse_bo_with_environmental_variables.
The steps for using this to obtain samples of a risk measure are as follows:
@@ -5480,7 +5494,7 @@ Utilities
[chen2018dpp]
-(1,2)
+(1,2)
Laming Chen and Guoxin Zhang and Hanning Zhou, Fast greedy MAP inference
for determinantal point process to improve recommendation diversity,
Proceedings of the 32nd International Conference on Neural Information
@@ -5530,7 +5544,7 @@
Utilities[burt2020svgp].
+of [burt2020svgp].
-
@@ -5538,7 +5552,7 @@
UtilitiesQualityFunction
A function measuring the quality of input points as their expected
improvement with respect to a conservative baseline. Expectations
-are according to the model from the previous BO step. See [moss2023ipa]
+are according to the model from the previous BO step. See [moss2023ipa]
for details and justification.
- Parameters:
@@ -5555,7 +5569,7 @@ Utilities
class botorch.models.utils.inducing_point_allocators.GreedyVarianceReduction[source]¶
Bases: InducingPointAllocator
-The inducing point allocator proposed by [burt2020svgp], that
+
The inducing point allocator proposed by [burt2020svgp], that
greedily chooses inducing point locations with maximal (conditional)
predictive variance.
@@ -5565,7 +5579,7 @@ UtilitiesInducingPointAllocator
An inducing point allocator that greedily chooses inducing points with large
predictive variance and that are in promising regions of the search
-space (according to the model form the previous BO step), see [moss2023ipa].
+space (according to the model form the previous BO step), see [moss2023ipa].
- Parameters:
@@ -5580,11 +5594,11 @@ Utilities
botorch.models.utils.inducing_point_allocators._pivoted_cholesky_init(train_inputs, kernel_matrix, max_length, quality_scores, epsilon=1e-06)[source]¶
A pivoted Cholesky initialization method for the inducing points,
-originally proposed in [burt2020svgp] with the algorithm itself coming from
-[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on
+originally proposed in [burt2020svgp] with the algorithm itself coming from
+[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on
https://github.com/laming-chen/fast-map-dpp/blob/master/dpp.py but with a small
modification to allow the underlying DPP to be defined through its diversity-quality
-decomposition,as discussed by [moss2023ipa]. This method returns a greedy
+decomposition,as discussed by [moss2023ipa]. This method returns a greedy
approximation of the MAP estimate of the specified DPP, i.e. its returns a
set of points that are highly diverse (according to the provided kernel_matrix)
and have high quality (according to the provided quality_scores).
diff --git a/v/latest/api/models/index.html b/v/latest/api/models/index.html
index e0582fedbd..f598b4e6ae 100644
--- a/v/latest/api/models/index.html
+++ b/v/latest/api/models/index.html
@@ -2347,18 +2347,31 @@ Models
Contextual GP Models with Context Rewards¶
+References
+
+
+[Feng2020HDCPS]
+Q. Feng, B. Latham, H. Mao and E. Backshy. High-Dimensional Contextual Policy
+Search with Unknown Context Rewards using Bayesian Optimization.
+Advances in Neural Information Processing Systems 33, NeurIPS 2020.
+
+
-
-class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
+class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, train_Yvar=None, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
Bases: MultiTaskGP
-The Multi-Task GP with the latent context embedding multioutput
-(LCE-M) kernel.
+The Multi-Task GP with the latent context embedding multioutput (LCE-M)
+kernel. See [Feng2020HDCPS] for a reference on the model and its use in Bayesian
+optimization.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
task_feature (int) – Column index of train_X to get context indices.
train_Yvar (Tensor | None) – An optional (n x 1) tensor of observed variances of each
+training Y. If None, we infer the noise. Note that the inferred noise
+is common across all tasks.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
number of categorical variables. If None, task indices will
@@ -2424,12 +2437,13 @@
ModelsBases: LCEMGP
The Multi-Task GP the latent context embedding multioutput
(LCE-M) kernel, with known observation noise.
+DEPRECATED: Please use LCEMGP with train_Yvar instead.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
train_Yvar (Tensor) – (n x 1) Noise variances of each training Y.
train_Yvar (Tensor) – (n x 1) Observed variances of each training Y.
task_feature (int) – Column index of train_X to get context indices.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context
features. Rows are ordered by context indices, where k is the
@@ -2461,21 +2475,21 @@
Models
[burt2020svgp]
-(1,2,3,4)
+(1,2,3,4)
David R. Burt and Carl Edward Rasmussen and Mark van der Wilk,
Convergence of Sparse Variational Inference in Gaussian Process Regression,
Journal of Machine Learning Research, 2020,
http://jmlr.org/papers/v21/19-1015.html.
-[hensman2013svgp]
+[hensman2013svgp]
James Hensman and Nicolo Fusi and Neil D. Lawrence, Gaussian Processes
for Big Data, Proceedings of the 29th Conference on Uncertainty in
Artificial Intelligence, 2013, https://arxiv.org/abs/1309.6835.
[moss2023ipa]
-(1,2,3,4)
+(1,2,3,4)
Henry B. Moss and Sebastian W. Ober and Victor Picheny,
Inducing Point Allocation for Sparse Gaussian Processes
in High-Throughput Bayesian Optimization,Proceedings of
@@ -2595,12 +2609,12 @@
Models
class botorch.models.approximate_gp.SingleTaskVariationalGP(train_X, train_Y=None, likelihood=None, num_outputs=1, learn_inducing_points=True, covar_module=None, mean_module=None, variational_distribution=None, variational_strategy=<class 'gpytorch.variational.variational_strategy.VariationalStrategy'>, inducing_points=None, outcome_transform=None, input_transform=None, inducing_point_allocator=None)[source]¶
Bases: ApproximateGPyTorchModel
-A single-task variational GP model following [hensman2013svgp].
+A single-task variational GP model following [hensman2013svgp].
By default, the inducing points are initialized though the
-GreedyVarianceReduction of [burt2020svgp], which is known to be
+GreedyVarianceReduction of [burt2020svgp], which is known to be
effective for building globally accurate models. However, custom
inducing point allocators designed for specific down-stream tasks can also be
-provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction
+provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction
when the goal is to build a model suitable for standard BO.
A single-task variational GP using relatively strong priors on the Kernel
hyperparameters, which work best when covariates are normalized to the unit
@@ -2712,7 +2726,7 @@
Models
[Eriksson2021saasbo]
-(1,2,3)
+(1,2,3)
D. Eriksson, M. Jankowiak. High-Dimensional Bayesian Optimization
with Sparse Axis-Aligned Subspaces. Proceedings of the Thirty-
Seventh Conference on Uncertainty in Artificial Intelligence, 2021.
@@ -2773,7 +2787,7 @@ ModelsImplementation of the sparse axis-aligned subspace priors (SAAS) model.
The SAAS model uses sparsity-inducing priors to identify the most important
parameters. This model is suitable for high-dimensional BO with potentially
-hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
+hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
SaasPyroModel is not a standard BoTorch model; instead, it is used as
an input to SaasFullyBayesianSingleTaskGP. It is used as a default keyword
argument, and end users are not likely to need to instantiate or modify a
@@ -2902,7 +2916,7 @@
ModelsThis model assumes that the inputs have been normalized to [0, 1]^d and that
the output has been standardized to have zero mean and unit variance. You can
either normalize and standardize the data before constructing the model or use
-an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo]
+an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo]
with a Matern-5/2 kernel is used by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it
isn’t compatible with fit_gpytorch_model.
@@ -3127,7 +3141,7 @@ ModelsA fully Bayesian multi-task GP model with the SAAS prior.
This model assumes that the inputs have been normalized to [0, 1]^d and that the
output has been stratified standardized to have zero mean and unit variance for
-each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data
+each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data
kernel by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it
isn’t compatible with fit_gpytorch_model.
@@ -3581,14 +3595,14 @@ Model Components
class botorch.models.kernels.orthogonal_additive_kernel.OrthogonalAdditiveKernel(base_kernel, dim, quad_deg=32, second_order=False, batch_shape=None, dtype=None, device=None, coeff_constraint=Positive())[source]¶
Bases: Kernel
-Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though
+
Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though
only for the case of Gaussian base kernels with a Gaussian input data distribution.
The implementation here generalizes OAKs to arbitrary base kernels by using a
Gauss-Legendre quadrature approximation to the required one-dimensional integrals
involving the base kernels.
-[Lu2022additive]
+[Lu2022additive]
X. Lu, A. Boukouvalas, and J. Hensman. Additive Gaussian processes revisited.
Proceedings of the 39th International Conference on Machine Learning. Jul 2022.
@@ -3818,7 +3832,7 @@ Transforms
-[eriksson2021scalable]
+[eriksson2021scalable]
D. Eriksson, M. Poloczek. Scalable Constrained Bayesian Optimization.
International Conference on Artificial Intelligence and Statistics. PMLR, 2021,
http://proceedings.mlr.press/v130/eriksson21a.html
@@ -4257,7 +4271,7 @@ Transformsclass botorch.models.transforms.outcome.Bilog(outputs=None)[source]¶
Bases: OutcomeTransform
Bilog-transform outcomes.
-The Bilog transform [eriksson2021scalable] is useful for modeling outcome
+
The Bilog transform [eriksson2021scalable] is useful for modeling outcome
constraints as it magnifies values near zero and flattens extreme values.
Bilog-transform outcomes.
@@ -4602,7 +4616,7 @@ Transforms[Daulton2022bopr]. This transformation supports differentiable
+proposed in [Daulton2022bopr]. This transformation supports differentiable
approximate rounding (currently only for integers). The rounding function
is approximated with a piece-wise function where each piece is a hyperbolic
tangent function.
@@ -4774,8 +4788,8 @@ Transforms[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition
-function introduced in [Cakmak2020risk] can be found at
+[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition
+function introduced in [Cakmak2020risk] can be found at
https://botorch.org/tutorials/risk_averse_bo_with_environmental_variables.
The steps for using this to obtain samples of a risk measure are as follows:
@@ -5480,7 +5494,7 @@ Utilities
[chen2018dpp]
-(1,2)
+(1,2)
Laming Chen and Guoxin Zhang and Hanning Zhou, Fast greedy MAP inference
for determinantal point process to improve recommendation diversity,
Proceedings of the 32nd International Conference on Neural Information
@@ -5530,7 +5544,7 @@
Utilities[burt2020svgp].
+of [burt2020svgp].
-
@@ -5538,7 +5552,7 @@
UtilitiesQualityFunction
A function measuring the quality of input points as their expected
improvement with respect to a conservative baseline. Expectations
-are according to the model from the previous BO step. See [moss2023ipa]
+are according to the model from the previous BO step. See [moss2023ipa]
for details and justification.
- Parameters:
@@ -5555,7 +5569,7 @@ Utilities
class botorch.models.utils.inducing_point_allocators.GreedyVarianceReduction[source]¶
Bases: InducingPointAllocator
-The inducing point allocator proposed by [burt2020svgp], that
+
The inducing point allocator proposed by [burt2020svgp], that
greedily chooses inducing point locations with maximal (conditional)
predictive variance.
@@ -5565,7 +5579,7 @@ UtilitiesInducingPointAllocator
An inducing point allocator that greedily chooses inducing points with large
predictive variance and that are in promising regions of the search
-space (according to the model form the previous BO step), see [moss2023ipa].
+space (according to the model form the previous BO step), see [moss2023ipa].
- Parameters:
@@ -5580,11 +5594,11 @@ Utilities
botorch.models.utils.inducing_point_allocators._pivoted_cholesky_init(train_inputs, kernel_matrix, max_length, quality_scores, epsilon=1e-06)[source]¶
A pivoted Cholesky initialization method for the inducing points,
-originally proposed in [burt2020svgp] with the algorithm itself coming from
-[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on
+originally proposed in [burt2020svgp] with the algorithm itself coming from
+[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on
https://github.com/laming-chen/fast-map-dpp/blob/master/dpp.py but with a small
modification to allow the underlying DPP to be defined through its diversity-quality
-decomposition,as discussed by [moss2023ipa]. This method returns a greedy
+decomposition,as discussed by [moss2023ipa]. This method returns a greedy
approximation of the MAP estimate of the specified DPP, i.e. its returns a
set of points that are highly diverse (according to the provided kernel_matrix)
and have high quality (according to the provided quality_scores).
diff --git a/v/latest/files/batch_mode_cross_validation.ipynb b/v/latest/files/batch_mode_cross_validation.ipynb
index eab9277ed9..383bcc8d3d 100644
--- a/v/latest/files/batch_mode_cross_validation.ipynb
+++ b/v/latest/files/batch_mode_cross_validation.ipynb
@@ -1,329 +1,330 @@
{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {
- "collapsed": true
- },
- "source": [
- "## Application of batch-mode regression to cross-validation\n",
- "\n",
- "botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Simple_GP_Regression/Simple_Batch_Mode_GP_Regression.ipynb) from GPyTorch:\n",
- "\n",
- "$$y = \\sin(2\\pi x) + \\epsilon, ~\\epsilon \\sim \\mathcal N(0, 0.2).$$\n",
- "\n",
- "Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see [ax.modelbridge.cross_validation](https://github.com/facebook/Ax/blob/main/ax/modelbridge/cross_validation.py). However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 1,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": [
- "import torch\n",
- "import math\n",
- "\n",
- "device = torch.device(\"cpu\")\n",
- "dtype = torch.float\n",
- "torch.manual_seed(3);"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Initialize the CV dataset\n",
- "\n",
- "For our training data, we take 20 regularly spaced points on the interval $[0, 1]$ and generate noisy evaluations with an observed noise variance of 0.2. Remember that botorch requires an explicit output dimension."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 2,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": [
- "sigma = math.sqrt(0.2)\n",
- "train_X = torch.linspace(0, 1, 20, dtype=dtype, device=device).view(-1, 1)\n",
- "train_Y_noiseless = torch.sin(train_X * (2 * math.pi))\n",
- "train_Y = train_Y_noiseless + sigma * torch.randn_like(train_Y_noiseless)\n",
- "train_Yvar = torch.full_like(train_Y, 0.2)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "The botorch function `gen_loo_cv_folds` takes our observed data `train_X`, `train_Y`, `train_Yvar` as input and returns the LOO CV folds in a `CVFolds` object."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 3,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": [
- "from botorch.cross_validation import gen_loo_cv_folds\n",
- "\n",
- "cv_folds = gen_loo_cv_folds(train_X=train_X, train_Y=train_Y, train_Yvar=train_Yvar)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "The `cv_folds` object contains the data, stored as tensors of appropriate batch shape, necessary to perform 20 CVs of 19 training points and 1 test point. For example, we can check that the shapes of the training inputs and training targets are `b x n x d = 20 x 19 x 1` and `b x n x o = 20 x 19 x 1` respectively, where `o` is the number of outputs."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 4,
- "metadata": {},
- "outputs": [
+ "cells": [
{
- "data": {
- "text/plain": [
- "(torch.Size([20, 20, 1]), torch.Size([20, 20, 1]))"
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": true
+ },
+ "source": [
+ "## Application of batch-mode regression to cross-validation\n",
+ "\n",
+ "botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Exact_GPs/Simple_GP_Regression.ipynb) from GPyTorch:\n",
+ "\n",
+ "$$y = \\sin(2\\pi x) + \\epsilon, ~\\epsilon \\sim \\mathcal N(0, 0.2).$$\n",
+ "\n",
+ "Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see [ax.modelbridge.cross_validation](https://github.com/facebook/Ax/blob/main/ax/modelbridge/cross_validation.py). However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial."
]
- },
- "execution_count": 4,
- "metadata": {
- "bento_obj_id": "140675275377288"
- },
- "output_type": "execute_result"
- }
- ],
- "source": [
- "cv_folds.train_X.shape, cv_folds.train_Y.shape"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 5,
- "metadata": {},
- "outputs": [
+ },
{
- "data": {
- "text/plain": [
- "(torch.Size([20, 1, 1]), torch.Size([20, 1, 1]))"
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "import torch\n",
+ "import math\n",
+ "\n",
+ "device = torch.device(\"cpu\")\n",
+ "dtype = torch.float\n",
+ "torch.manual_seed(3);"
]
- },
- "execution_count": 5,
- "metadata": {
- "bento_obj_id": "140676807470216"
- },
- "output_type": "execute_result"
- }
- ],
- "source": [
- "cv_folds.test_X.shape, cv_folds.test_Y.shape"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Note that in a situation where the dataset is large, one may not want to perform LOO; in that case, a similar process can be used to perform $k$-fold CV."
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Perform LOOCV\n",
- "\n",
- "We can use the `batch_cross_validation` function to perform LOOCV using batching (meaning that the `b = 20` sets of training data can be fit as `b = 20` separate GP models with separate hyperparameters in parallel through GPyTorch) and return a CVResult tuple with the batched `GPyTorchPosterior` object over the LOOCV test points and the observed targets. The `batch_cross_validation` requires a model class (`model_cls`) and a marginal log likelihood class (`mll_cls`). Since we have an observed and constant noise level, we will use the FixedNoiseGP as the `model_cls` and an ExactMarginalLogLikelihood as the `mll_cls`."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 6,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": [
- "from botorch.cross_validation import batch_cross_validation\n",
- "from botorch.models import FixedNoiseGP\n",
- "from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood\n",
- "\n",
- "# instantiate and fit model\n",
- "cv_results = batch_cross_validation(\n",
- " model_cls=FixedNoiseGP,\n",
- " mll_cls=ExactMarginalLogLikelihood,\n",
- " cv_folds=cv_folds,\n",
- ")"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "#### Compute the cross-validation error and generate plots\n",
- "To compute the cross-validation error, we first evaluate the test points by computing the posterior in batch mode. Next, we compute the squared errors for each test point from the prediction and take an average across all cross-validation folds."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [
+ },
{
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "Cross-validation error: 0.062\n"
- ]
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Initialize the CV dataset\n",
+ "\n",
+ "For our training data, we take 20 regularly spaced points on the interval $[0, 1]$ and generate noisy evaluations with an observed noise variance of 0.2. Remember that botorch requires an explicit output dimension."
+ ]
},
{
- "data": {
- "image/png": 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RCU6g5HC8M49zxV5DW53/fUBCmGKTGFN+sD1/N/a8vDwGDhxIVlYWSUlJpKen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\niFSi5CAiIpUoOYiISCWeHJU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- "text/plain": [
- ""
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "sigma = math.sqrt(0.2)\n",
+ "train_X = torch.linspace(0, 1, 20, dtype=dtype, device=device).view(-1, 1)\n",
+ "train_Y_noiseless = torch.sin(train_X * (2 * math.pi))\n",
+ "train_Y = train_Y_noiseless + sigma * torch.randn_like(train_Y_noiseless)\n",
+ "train_Yvar = torch.full_like(train_Y, 0.2)"
]
- },
- "metadata": {
- "bento_obj_id": "140674656098008"
- },
- "output_type": "display_data"
- }
- ],
- "source": [
- "from matplotlib import pyplot as plt\n",
- "\n",
- "%matplotlib inline\n",
- "\n",
- "posterior = cv_results.posterior\n",
- "mean = posterior.mean\n",
- "cv_error = ((cv_folds.test_Y.squeeze() - mean.squeeze()) ** 2).mean()\n",
- "print(f\"Cross-validation error: {cv_error : 4.2}\")\n",
- "\n",
- "# get lower and upper confidence bounds\n",
- "lower, upper = posterior.mvn.confidence_region()\n",
- "\n",
- "# scatterplot of predicted versus test\n",
- "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
- "plt.plot([-1.5, 1.5], [-1.5, 1.5], \"k\", label=\"true objective\", linewidth=2)\n",
- "\n",
- "axes.set_xlabel(\"Actual\")\n",
- "axes.set_ylabel(\"Predicted\")\n",
- "\n",
- "axes.errorbar(\n",
- " x=cv_folds.test_Y.numpy().flatten(),\n",
- " y=mean.numpy().flatten(),\n",
- " xerr=1.96 * sigma,\n",
- " yerr=((upper - lower) / 2).numpy().flatten(),\n",
- " fmt=\"*\",\n",
- ");"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Finally, we can visualize the fitted models. To do this, we again take advantage of batch-mode evaluation to obtain predictions, including lower and upper confidence regions, from each of the 20 models."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 8,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": [
- "model = cv_results.model\n",
- "with torch.no_grad():\n",
- " # evaluate the models at a series of points for plotting\n",
- " plot_x = (\n",
- " torch.linspace(0, 1, 101).view(1, -1, 1).repeat(cv_folds.train_X.shape[0], 1, 1)\n",
- " )\n",
- " posterior = model.posterior(plot_x)\n",
- " mean = posterior.mean\n",
- "\n",
- " # get lower and upper confidence bounds\n",
- " lower, upper = posterior.mvn.confidence_region()\n",
- " plot_x.squeeze_()"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "The code snippet below plots the result for the 12th CV fold (by setting `num = 12`), but note that we have computed the results for all folds above (other plots can be obtained by iterating `num` from 1 to 20)."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 9,
- "metadata": {},
- "outputs": [
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The botorch function `gen_loo_cv_folds` takes our observed data `train_X`, `train_Y`, `train_Yvar` as input and returns the LOO CV folds in a `CVFolds` object."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "from botorch.cross_validation import gen_loo_cv_folds\n",
+ "\n",
+ "cv_folds = gen_loo_cv_folds(train_X=train_X, train_Y=train_Y, train_Yvar=train_Yvar)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The `cv_folds` object contains the data, stored as tensors of appropriate batch shape, necessary to perform 20 CVs of 19 training points and 1 test point. For example, we can check that the shapes of the training inputs and training targets are `b x n x d = 20 x 19 x 1` and `b x n x o = 20 x 19 x 1` respectively, where `o` is the number of outputs."
+ ]
+ },
{
- "data": {
- "image/png": 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CzJDPBx/tqeRQtTp9iQRlVDTY2VfZpHUY3dLnk7vPB/LfM8jId3Lmj5T/qDU8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\nxVG+OGylxqZOb50am4MPdpb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FUSRu13e/v25fK3kDXAwY1coFP7GSnufsdMVsIEsZeyI7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\n3AVBEMKQSO6CIAhhSCR3QRCEMPT/J+miawYbwJIAAAAASUVORK5CYII=\n",
- "text/plain": [
- ""
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(torch.Size([20, 20, 1]), torch.Size([20, 20, 1]))"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {
+ "bento_obj_id": "140675275377288"
+ },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "cv_folds.train_X.shape, cv_folds.train_Y.shape"
]
- },
- "metadata": {
- "bento_obj_id": "140674628290824"
- },
- "output_type": "display_data"
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(torch.Size([20, 1, 1]), torch.Size([20, 1, 1]))"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {
+ "bento_obj_id": "140676807470216"
+ },
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "cv_folds.test_X.shape, cv_folds.test_Y.shape"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Note that in a situation where the dataset is large, one may not want to perform LOO; in that case, a similar process can be used to perform $k$-fold CV."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Perform LOOCV\n",
+ "\n",
+ "We can use the `batch_cross_validation` function to perform LOOCV using batching (meaning that the `b = 20` sets of training data can be fit as `b = 20` separate GP models with separate hyperparameters in parallel through GPyTorch) and return a CVResult tuple with the batched `GPyTorchPosterior` object over the LOOCV test points and the observed targets. The `batch_cross_validation` requires a model class (`model_cls`) and a marginal log likelihood class (`mll_cls`). Since we have an observed and constant noise level, we will use the FixedNoiseGP as the `model_cls` and an ExactMarginalLogLikelihood as the `mll_cls`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "from botorch.cross_validation import batch_cross_validation\n",
+ "from botorch.models import FixedNoiseGP\n",
+ "from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood\n",
+ "\n",
+ "# instantiate and fit model\n",
+ "cv_results = batch_cross_validation(\n",
+ " model_cls=FixedNoiseGP,\n",
+ " mll_cls=ExactMarginalLogLikelihood,\n",
+ " cv_folds=cv_folds,\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "#### Compute the cross-validation error and generate plots\n",
+ "To compute the cross-validation error, we first evaluate the test points by computing the posterior in batch mode. Next, we compute the squared errors for each test point from the prediction and take an average across all cross-validation folds."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Cross-validation error: 0.062\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "bento_obj_id": "140674656098008"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "from matplotlib import pyplot as plt\n",
+ "\n",
+ "%matplotlib inline\n",
+ "\n",
+ "posterior = cv_results.posterior\n",
+ "mean = posterior.mean\n",
+ "cv_error = ((cv_folds.test_Y.squeeze() - mean.squeeze()) ** 2).mean()\n",
+ "print(f\"Cross-validation error: {cv_error : 4.2}\")\n",
+ "\n",
+ "# get lower and upper confidence bounds\n",
+ "lower, upper = posterior.mvn.confidence_region()\n",
+ "\n",
+ "# scatterplot of predicted versus test\n",
+ "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
+ "plt.plot([-1.5, 1.5], [-1.5, 1.5], \"k\", label=\"true objective\", linewidth=2)\n",
+ "\n",
+ "axes.set_xlabel(\"Actual\")\n",
+ "axes.set_ylabel(\"Predicted\")\n",
+ "\n",
+ "axes.errorbar(\n",
+ " x=cv_folds.test_Y.numpy().flatten(),\n",
+ " y=mean.numpy().flatten(),\n",
+ " xerr=1.96 * sigma,\n",
+ " yerr=((upper - lower) / 2).numpy().flatten(),\n",
+ " fmt=\"*\",\n",
+ ");"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finally, we can visualize the fitted models. To do this, we again take advantage of batch-mode evaluation to obtain predictions, including lower and upper confidence regions, from each of the 20 models."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "model = cv_results.model\n",
+ "with torch.no_grad():\n",
+ " # evaluate the models at a series of points for plotting\n",
+ " plot_x = (\n",
+ " torch.linspace(0, 1, 101).view(1, -1, 1).repeat(cv_folds.train_X.shape[0], 1, 1)\n",
+ " )\n",
+ " posterior = model.posterior(plot_x)\n",
+ " mean = posterior.mean\n",
+ "\n",
+ " # get lower and upper confidence bounds\n",
+ " lower, upper = posterior.mvn.confidence_region()\n",
+ " plot_x.squeeze_()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The code snippet below plots the result for the 12th CV fold (by setting `num = 12`), but note that we have computed the results for all folds above (other plots can be obtained by iterating `num` from 1 to 20)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "bento_obj_id": "140674628290824"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
+ "\n",
+ "# plot the 12th CV fold\n",
+ "num = 12\n",
+ "\n",
+ "# plot the training data in black\n",
+ "axes.plot(\n",
+ " cv_folds.train_X[num - 1].detach().numpy(),\n",
+ " cv_folds.train_Y[num - 1].detach().numpy(),\n",
+ " \"k*\",\n",
+ ")\n",
+ "\n",
+ "# plot the test data in red\n",
+ "axes.plot(\n",
+ " cv_folds.test_X[num - 1].detach().numpy(),\n",
+ " cv_folds.test_Y[num - 1].detach().numpy(),\n",
+ " \"r*\",\n",
+ ")\n",
+ "\n",
+ "# plot posterior means as blue line\n",
+ "axes.plot(plot_x[num - 1].numpy(), mean[num - 1].numpy(), \"b\")\n",
+ "\n",
+ "# shade between the lower and upper confidence bounds\n",
+ "axes.fill_between(\n",
+ " plot_x[num - 1].numpy(), lower[num - 1].numpy(), upper[num - 1].numpy(), alpha=0.5\n",
+ ");"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "fileHeader": "",
+ "kernelspec": {
+ "display_name": "python3",
+ "language": "python",
+ "name": "python3"
}
- ],
- "source": [
- "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
- "\n",
- "# plot the 12th CV fold\n",
- "num = 12\n",
- "\n",
- "# plot the training data in black\n",
- "axes.plot(\n",
- " cv_folds.train_X[num - 1].detach().numpy(),\n",
- " cv_folds.train_Y[num - 1].detach().numpy(),\n",
- " \"k*\",\n",
- ")\n",
- "\n",
- "# plot the test data in red\n",
- "axes.plot(\n",
- " cv_folds.test_X[num - 1].detach().numpy(),\n",
- " cv_folds.test_Y[num - 1].detach().numpy(),\n",
- " \"r*\",\n",
- ")\n",
- "\n",
- "# plot posterior means as blue line\n",
- "axes.plot(plot_x[num - 1].numpy(), mean[num - 1].numpy(), \"b\")\n",
- "\n",
- "# shade between the lower and upper confidence bounds\n",
- "axes.fill_between(\n",
- " plot_x[num - 1].numpy(), lower[num - 1].numpy(), upper[num - 1].numpy(), alpha=0.5\n",
- ");"
- ]
},
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": true
- },
- "outputs": [],
- "source": []
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "python3",
- "language": "python",
- "name": "python3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat": 4,
+ "nbformat_minor": 2
}
diff --git a/v/latest/files/batch_mode_cross_validation.py b/v/latest/files/batch_mode_cross_validation.py
index f87343a3f0..e6ce037214 100644
--- a/v/latest/files/batch_mode_cross_validation.py
+++ b/v/latest/files/batch_mode_cross_validation.py
@@ -3,7 +3,7 @@
# ## Application of batch-mode regression to cross-validation
#
-# botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Simple_GP_Regression/Simple_Batch_Mode_GP_Regression.ipynb) from GPyTorch:
+# botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Exact_GPs/Simple_GP_Regression.ipynb) from GPyTorch:
#
# $$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$
#
diff --git a/v/latest/js/searchindex.js b/v/latest/js/searchindex.js
index 35689eb4be..92893868cb 100644
--- a/v/latest/js/searchindex.js
+++ b/v/latest/js/searchindex.js
@@ -1 +1 @@
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"botorch.utils.transforms.normalize_indices"]], "normalize_linear_constraints() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.normalize_linear_constraints"]], "num_outputs (botorch.utils.testing.mockmodel property)": [[13, "botorch.utils.testing.MockModel.num_outputs"]], "optimize_posterior_samples() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.optimize_posterior_samples"]], "parameter_rollback_ctx() (in module botorch.utils.context_managers)": [[13, "botorch.utils.context_managers.parameter_rollback_ctx"]], "pareto_y (botorch.utils.multi_objective.box_decompositions.box_decomposition_list.boxdecompositionlist property)": [[13, "botorch.utils.multi_objective.box_decompositions.box_decomposition_list.BoxDecompositionList.pareto_Y"]], "perm (botorch.utils.probability.linalg.pivotedcholesky attribute)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.perm"]], "perm (botorch.utils.probability.mvnxpb.mvnxpbstate attribute)": [[13, "botorch.utils.probability.mvnxpb.mvnxpbState.perm"]], "phi() (in module botorch.utils.probability.utils)": [[13, "botorch.utils.probability.utils.phi"]], "piv_chol (botorch.utils.probability.mvnxpb.mvnxpbstate attribute)": [[13, "botorch.utils.probability.mvnxpb.mvnxpbState.piv_chol"]], "pivot_() (botorch.utils.probability.linalg.pivotedcholesky method)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.pivot_"]], "pivot_() (botorch.utils.probability.mvnxpb.mvnxpb method)": [[13, "botorch.utils.probability.mvnxpb.MVNXPB.pivot_"]], "plug_ins (botorch.utils.probability.mvnxpb.mvnxpbstate attribute)": [[13, "botorch.utils.probability.mvnxpb.mvnxpbState.plug_ins"]], "pop() (botorch.utils.torch.bufferdict method)": [[13, "botorch.utils.torch.BufferDict.pop"]], "posterior() (botorch.utils.testing.mockmodel method)": [[13, "botorch.utils.testing.MockModel.posterior"]], "ref_point (botorch.utils.multi_objective.box_decompositions.box_decomposition_list.boxdecompositionlist property)": [[13, "botorch.utils.multi_objective.box_decompositions.box_decomposition_list.BoxDecompositionList.ref_point"]], "ref_point (botorch.utils.multi_objective.hypervolume.hypervolume property)": [[13, "botorch.utils.multi_objective.hypervolume.Hypervolume.ref_point"]], "reinsert() (botorch.utils.multi_objective.hypervolume.multilist method)": [[13, "botorch.utils.multi_objective.hypervolume.MultiList.reinsert"]], "remove() (botorch.utils.multi_objective.hypervolume.multilist method)": [[13, "botorch.utils.multi_objective.hypervolume.MultiList.remove"]], "requires_grad_ctx() (in module botorch.utils.context_managers)": [[13, "botorch.utils.context_managers.requires_grad_ctx"]], "rsample() (botorch.utils.probability.truncated_multivariate_normal.truncatedmultivariatenormal method)": [[13, "botorch.utils.probability.truncated_multivariate_normal.TruncatedMultivariateNormal.rsample"]], "rsample() (botorch.utils.probability.unified_skew_normal.unifiedskewnormal method)": [[13, "botorch.utils.probability.unified_skew_normal.UnifiedSkewNormal.rsample"]], "rsample() (botorch.utils.testing.mockposterior method)": [[13, "botorch.utils.testing.MockPosterior.rsample"]], "rsample_from_base_samples() (botorch.utils.testing.mockposterior method)": [[13, "botorch.utils.testing.MockPosterior.rsample_from_base_samples"]], "sample_cached_cholesky() (in module botorch.utils.low_rank)": [[13, "botorch.utils.low_rank.sample_cached_cholesky"]], "sample_hypersphere() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.sample_hypersphere"]], "sample_polytope() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.sample_polytope"]], "sample_simplex() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.sample_simplex"]], "sampler (botorch.utils.probability.truncated_multivariate_normal.truncatedmultivariatenormal property)": [[13, "botorch.utils.probability.truncated_multivariate_normal.TruncatedMultivariateNormal.sampler"]], "scale_tril (botorch.utils.probability.unified_skew_normal.unifiedskewnormal property)": [[13, "botorch.utils.probability.unified_skew_normal.UnifiedSkewNormal.scale_tril"]], "select_pivot() (botorch.utils.probability.mvnxpb.mvnxpb method)": [[13, "botorch.utils.probability.mvnxpb.MVNXPB.select_pivot"]], "setup() (botorch.utils.testing.botorchtestcase method)": [[13, "botorch.utils.testing.BotorchTestCase.setUp"]], "set_x_pending() (botorch.utils.testing.mockacquisitionfunction method)": [[13, "botorch.utils.testing.MockAcquisitionFunction.set_X_pending"]], "shape (botorch.utils.containers.densecontainer property)": [[13, "botorch.utils.containers.DenseContainer.shape"]], "shape (botorch.utils.containers.slicecontainer property)": [[13, "botorch.utils.containers.SliceContainer.shape"]], "sigmoid() (in module botorch.utils.safe_math)": [[13, "botorch.utils.safe_math.sigmoid"]], "smooth_amax() (in module botorch.utils.safe_math)": [[13, "botorch.utils.safe_math.smooth_amax"]], "soft_eval_constraint() (in module botorch.utils.objective)": [[13, "botorch.utils.objective.soft_eval_constraint"]], "solve() (botorch.utils.probability.mvnxpb.mvnxpb method)": [[13, "botorch.utils.probability.mvnxpb.MVNXPB.solve"]], "solver (botorch.utils.probability.truncated_multivariate_normal.truncatedmultivariatenormal property)": [[13, "botorch.utils.probability.truncated_multivariate_normal.TruncatedMultivariateNormal.solver"]], "sort_by_dimension() (in module botorch.utils.multi_objective.hypervolume)": [[13, "botorch.utils.multi_objective.hypervolume.sort_by_dimension"]], "source() (botorch.utils.dispatcher.dispatcher method)": [[13, "botorch.utils.dispatcher.Dispatcher.source"]], "sparse_to_dense_constraints() (in module botorch.utils.sampling)": [[13, "botorch.utils.sampling.sparse_to_dense_constraints"]], "squeeze_last_dim() (in module botorch.utils.transforms)": [[13, "botorch.utils.transforms.squeeze_last_dim"]], "standard_normal_log_hazard() (in module botorch.utils.probability.utils)": [[13, "botorch.utils.probability.utils.standard_normal_log_hazard"]], "standardize() (in module botorch.utils.transforms)": [[13, "botorch.utils.transforms.standardize"]], "state_dict() (botorch.utils.testing.mockmodel method)": [[13, "botorch.utils.testing.MockModel.state_dict"]], "step (botorch.utils.probability.linalg.pivotedcholesky attribute)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.step"]], "step (botorch.utils.probability.mvnxpb.mvnxpbstate attribute)": [[13, "botorch.utils.probability.mvnxpb.mvnxpbState.step"]], "step() (botorch.utils.probability.lin_ess.linearellipticalslicesampler method)": [[13, "botorch.utils.probability.lin_ess.LinearEllipticalSliceSampler.step"]], "sub() (in module botorch.utils.safe_math)": [[13, "botorch.utils.safe_math.sub"]], "swap_along_dim_() (in module botorch.utils.probability.utils)": [[13, "botorch.utils.probability.utils.swap_along_dim_"]], "t_batch_mode_transform() (in module botorch.utils.transforms)": [[13, "botorch.utils.transforms.t_batch_mode_transform"]], "test_attributes() (botorch.utils.testing.multiobjectivetestproblemtestcasemixin method)": [[13, "botorch.utils.testing.MultiObjectiveTestProblemTestCaseMixin.test_attributes"]], "test_evaluate_slack_true() (botorch.utils.testing.constrainedtestproblemtestcasemixin method)": [[13, "botorch.utils.testing.ConstrainedTestProblemTestCaseMixin.test_evaluate_slack_true"]], "test_forward() (botorch.utils.testing.basetestproblemtestcasemixin method)": [[13, "botorch.utils.testing.BaseTestProblemTestCaseMixIn.test_forward"]], "test_max_hv() (botorch.utils.testing.multiobjectivetestproblemtestcasemixin method)": [[13, "botorch.utils.testing.MultiObjectiveTestProblemTestCaseMixin.test_max_hv"]], "test_num_constraints() (botorch.utils.testing.constrainedtestproblemtestcasemixin method)": [[13, "botorch.utils.testing.ConstrainedTestProblemTestCaseMixin.test_num_constraints"]], "test_optimal_value() (botorch.utils.testing.synthetictestfunctiontestcasemixin method)": [[13, "botorch.utils.testing.SyntheticTestFunctionTestCaseMixin.test_optimal_value"]], "test_optimizer() (botorch.utils.testing.synthetictestfunctiontestcasemixin method)": [[13, "botorch.utils.testing.SyntheticTestFunctionTestCaseMixin.test_optimizer"]], "test_ref_point() (botorch.utils.testing.multiobjectivetestproblemtestcasemixin method)": [[13, "botorch.utils.testing.MultiObjectiveTestProblemTestCaseMixin.test_ref_point"]], "training (botorch.utils.gp_sampling.gpdraw attribute)": [[13, "botorch.utils.gp_sampling.GPDraw.training"]], "training (botorch.utils.gp_sampling.randomfourierfeatures attribute)": [[13, "botorch.utils.gp_sampling.RandomFourierFeatures.training"]], "training (botorch.utils.multi_objective.box_decompositions.box_decomposition_list.boxdecompositionlist attribute)": [[13, "botorch.utils.multi_objective.box_decompositions.box_decomposition_list.BoxDecompositionList.training"]], "training (botorch.utils.multi_objective.box_decompositions.dominated.dominatedpartitioning attribute)": [[13, "botorch.utils.multi_objective.box_decompositions.dominated.DominatedPartitioning.training"]], "training (botorch.utils.multi_objective.box_decompositions.non_dominated.fastnondominatedpartitioning attribute)": [[13, "botorch.utils.multi_objective.box_decompositions.non_dominated.FastNondominatedPartitioning.training"]], "training (botorch.utils.multi_objective.box_decompositions.non_dominated.nondominatedpartitioning attribute)": [[13, "botorch.utils.multi_objective.box_decompositions.non_dominated.NondominatedPartitioning.training"]], "training (botorch.utils.torch.bufferdict attribute)": [[13, "botorch.utils.torch.BufferDict.training"]], "tril (botorch.utils.probability.linalg.pivotedcholesky attribute)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.tril"]], "type_bypassing_encoder() (in module botorch.utils.dispatcher)": [[13, "botorch.utils.dispatcher.type_bypassing_encoder"]], "unnormalize() (in module botorch.utils.transforms)": [[13, "botorch.utils.transforms.unnormalize"]], "update() (botorch.utils.multi_objective.box_decompositions.box_decomposition_list.boxdecompositionlist method)": [[13, "botorch.utils.multi_objective.box_decompositions.box_decomposition_list.BoxDecompositionList.update"]], "update() (botorch.utils.torch.bufferdict method)": [[13, "botorch.utils.torch.BufferDict.update"]], "update_() (botorch.utils.probability.linalg.pivotedcholesky method)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.update_"]], "update_local_upper_bounds_incremental() (in module botorch.utils.multi_objective.box_decompositions.utils)": [[13, "botorch.utils.multi_objective.box_decompositions.utils.update_local_upper_bounds_incremental"]], "validate_init (botorch.utils.probability.linalg.pivotedcholesky attribute)": [[13, "botorch.utils.probability.linalg.PivotedCholesky.validate_init"]], "values 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\ No newline at end of file
diff --git a/v/latest/tutorials/batch_mode_cross_validation.html b/v/latest/tutorials/batch_mode_cross_validation.html
index 4c92ff0202..786f41e9b7 100644
--- a/v/latest/tutorials/batch_mode_cross_validation.html
+++ b/v/latest/tutorials/batch_mode_cross_validation.html
@@ -72,7 +72,7 @@
-Application of batch-mode regression to cross-validation¶
botorch provides a helper function gen_loo_cv_folds to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:
+Application of batch-mode regression to cross-validation¶
botorch provides a helper function gen_loo_cv_folds to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:
$$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$
Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see ax.modelbridge.cross_validation. However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial.
@@ -273,218 +273,7 @@ Compute the cross
-
+
@@ -566,313 +355,7 @@ Compute the cross
-
+
diff --git a/v/latest/tutorials/batch_mode_cross_validation/index.html b/v/latest/tutorials/batch_mode_cross_validation/index.html
index 4c92ff0202..786f41e9b7 100644
--- a/v/latest/tutorials/batch_mode_cross_validation/index.html
+++ b/v/latest/tutorials/batch_mode_cross_validation/index.html
@@ -72,7 +72,7 @@
-Application of batch-mode regression to cross-validation¶
botorch provides a helper function gen_loo_cv_folds to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:
+Application of batch-mode regression to cross-validation¶
botorch provides a helper function gen_loo_cv_folds to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:
$$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$
Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see ax.modelbridge.cross_validation. However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial.
@@ -273,218 +273,7 @@ Compute the cross
-
+
@@ -566,313 +355,7 @@ Compute the cross
-
+
Contextual GP Models with Context Rewards¶
+References
+
+
+[Feng2020HDCPS]
+
+Q. Feng, B. Latham, H. Mao and E. Backshy. High-Dimensional Contextual Policy +Search with Unknown Context Rewards using Bayesian Optimization. +Advances in Neural Information Processing Systems 33, NeurIPS 2020.
+- -class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶ +class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, train_Yvar=None, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
Bases:
-MultiTaskGPThe Multi-Task GP with the latent context embedding multioutput -(LCE-M) kernel.
+The Multi-Task GP with the latent context embedding multioutput (LCE-M) +kernel. See [Feng2020HDCPS] for a reference on the model and its use in Bayesian +optimization.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
task_feature (int) – Column index of train_X to get context indices.
+train_Yvar (Tensor | None) – An optional (n x 1) tensor of observed variances of each +training Y. If None, we infer the noise. Note that the inferred noise +is common across all tasks.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context features. Rows are ordered by context indices, where k is the number of categorical variables. If None, task indices will @@ -2424,12 +2437,13 @@
Models
Bases:
LCEMGPThe Multi-Task GP the latent context embedding multioutput (LCE-M) kernel, with known observation noise.
+DEPRECATED: Please use LCEMGP with train_Yvar instead.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
-train_Yvar (Tensor) – (n x 1) Noise variances of each training Y.
+train_Yvar (Tensor) – (n x 1) Observed variances of each training Y.
task_feature (int) – Column index of train_X to get context indices.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context features. Rows are ordered by context indices, where k is the @@ -2461,21 +2475,21 @@
Models
[burt2020svgp] -(1,2,3,4) +(1,2,3,4)David R. Burt and Carl Edward Rasmussen and Mark van der Wilk, Convergence of Sparse Variational Inference in Gaussian Process Regression, Journal of Machine Learning Research, 2020, http://jmlr.org/papers/v21/19-1015.html.
-[hensman2013svgp] +[hensman2013svgp]James Hensman and Nicolo Fusi and Neil D. Lawrence, Gaussian Processes for Big Data, Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, 2013, https://arxiv.org/abs/1309.6835.
[moss2023ipa] -(1,2,3,4) +(1,2,3,4)Henry B. Moss and Sebastian W. Ober and Victor Picheny, Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimization,Proceedings of @@ -2595,12 +2609,12 @@
Models class botorch.models.approximate_gp.SingleTaskVariationalGP(train_X, train_Y=None, likelihood=None, num_outputs=1, learn_inducing_points=True, covar_module=None, mean_module=None, variational_distribution=None, variational_strategy=<class 'gpytorch.variational.variational_strategy.VariationalStrategy'>, inducing_points=None, outcome_transform=None, input_transform=None, inducing_point_allocator=None)[source]¶
Bases:
-ApproximateGPyTorchModelA single-task variational GP model following [hensman2013svgp].
+A single-task variational GP model following [hensman2013svgp].
By default, the inducing points are initialized though the -GreedyVarianceReduction of [burt2020svgp], which is known to be +GreedyVarianceReduction of [burt2020svgp], which is known to be effective for building globally accurate models. However, custom inducing point allocators designed for specific down-stream tasks can also be -provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction +provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction when the goal is to build a model suitable for standard BO.
A single-task variational GP using relatively strong priors on the Kernel hyperparameters, which work best when covariates are normalized to the unit @@ -2712,7 +2726,7 @@
Models
[Eriksson2021saasbo] -(1,2,3) +(1,2,3)D. Eriksson, M. Jankowiak. High-Dimensional Bayesian Optimization with Sparse Axis-Aligned Subspaces. Proceedings of the Thirty- Seventh Conference on Uncertainty in Artificial Intelligence, 2021.
@@ -2773,7 +2787,7 @@ModelsImplementation of the sparse axis-aligned subspace priors (SAAS) model.
The SAAS model uses sparsity-inducing priors to identify the most important parameters. This model is suitable for high-dimensional BO with potentially -hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
+hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.SaasPyroModel is not a standard BoTorch model; instead, it is used as an input to SaasFullyBayesianSingleTaskGP. It is used as a default keyword argument, and end users are not likely to need to instantiate or modify a @@ -2902,7 +2916,7 @@
ModelsThis model assumes that the inputs have been normalized to [0, 1]^d and that the output has been standardized to have zero mean and unit variance. You can either normalize and standardize the data before constructing the model or use -an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo] +an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo] with a Matern-5/2 kernel is used by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it isn’t compatible with fit_gpytorch_model.
@@ -3127,7 +3141,7 @@ModelsA fully Bayesian multi-task GP model with the SAAS prior.
This model assumes that the inputs have been normalized to [0, 1]^d and that the output has been stratified standardized to have zero mean and unit variance for -each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data +each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data kernel by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it isn’t compatible with fit_gpytorch_model.
@@ -3581,14 +3595,14 @@Model Components class botorch.models.kernels.orthogonal_additive_kernel.OrthogonalAdditiveKernel(base_kernel, dim, quad_deg=32, second_order=False, batch_shape=None, dtype=None, device=None, coeff_constraint=Positive())[source]¶
Bases:
-KernelOrthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though +
Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though only for the case of Gaussian base kernels with a Gaussian input data distribution.
The implementation here generalizes OAKs to arbitrary base kernels by using a Gauss-Legendre quadrature approximation to the required one-dimensional integrals involving the base kernels.
-[Lu2022additive] +[Lu2022additive]@@ -3818,7 +3832,7 @@X. Lu, A. Boukouvalas, and J. Hensman. Additive Gaussian processes revisited. Proceedings of the 39th International Conference on Machine Learning. Jul 2022.
Transforms
-[eriksson2021scalable] +[eriksson2021scalable]D. Eriksson, M. Poloczek. Scalable Constrained Bayesian Optimization. International Conference on Artificial Intelligence and Statistics. PMLR, 2021, http://proceedings.mlr.press/v130/eriksson21a.html
@@ -4257,7 +4271,7 @@Transformsclass botorch.models.transforms.outcome.Bilog(outputs=None)[source]¶
Bases:
OutcomeTransformBilog-transform outcomes.
-The Bilog transform [eriksson2021scalable] is useful for modeling outcome +
The Bilog transform [eriksson2021scalable] is useful for modeling outcome constraints as it magnifies values near zero and flattens extreme values.
Bilog-transform outcomes.
-
@@ -4602,7 +4616,7 @@
-
@@ -5538,7 +5552,7 @@
Utilities
QualityFunctionA function measuring the quality of input points as their expected improvement with respect to a conservative baseline. Expectations -are according to the model from the previous BO step. See [moss2023ipa] +are according to the model from the previous BO step. See [moss2023ipa] for details and justification.
- Parameters: @@ -5555,7 +5569,7 @@
Bases:
-InducingPointAllocatorThe inducing point allocator proposed by [burt2020svgp], that +
The inducing point allocator proposed by [burt2020svgp], that greedily chooses inducing point locations with maximal (conditional) predictive variance.
- Parameters:
-
@@ -5580,11 +5594,11 @@
A pivoted Cholesky initialization method for the inducing points, -originally proposed in [burt2020svgp] with the algorithm itself coming from -[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on +originally proposed in [burt2020svgp] with the algorithm itself coming from +[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on https://github.com/laming-chen/fast-map-dpp/blob/master/dpp.py but with a small modification to allow the underlying DPP to be defined through its diversity-quality -decomposition,as discussed by [moss2023ipa]. This method returns a greedy +decomposition,as discussed by [moss2023ipa]. This method returns a greedy approximation of the MAP estimate of the specified DPP, i.e. its returns a set of points that are highly diverse (according to the provided kernel_matrix) and have high quality (according to the provided quality_scores).
diff --git a/v/latest/api/models/index.html b/v/latest/api/models/index.html index e0582fedbd..f598b4e6ae 100644 --- a/v/latest/api/models/index.html +++ b/v/latest/api/models/index.html @@ -2347,18 +2347,31 @@Models
Contextual GP Models with Context Rewards¶
+References
+++[Feng2020HDCPS] ++Q. Feng, B. Latham, H. Mao and E. Backshy. High-Dimensional Contextual Policy +Search with Unknown Context Rewards using Bayesian Optimization. +Advances in Neural Information Processing Systems 33, NeurIPS 2020.
+- -class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶ +class botorch.models.contextual_multioutput.LCEMGP(train_X, train_Y, task_feature, train_Yvar=None, context_cat_feature=None, context_emb_feature=None, embs_dim_list=None, output_tasks=None, input_transform=None, outcome_transform=None)[source]¶
Bases:
-MultiTaskGPThe Multi-Task GP with the latent context embedding multioutput -(LCE-M) kernel.
+The Multi-Task GP with the latent context embedding multioutput (LCE-M) +kernel. See [Feng2020HDCPS] for a reference on the model and its use in Bayesian +optimization.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
task_feature (int) – Column index of train_X to get context indices.
+train_Yvar (Tensor | None) – An optional (n x 1) tensor of observed variances of each +training Y. If None, we infer the noise. Note that the inferred noise +is common across all tasks.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context features. Rows are ordered by context indices, where k is the number of categorical variables. If None, task indices will @@ -2424,12 +2437,13 @@
Models
Bases:
LCEMGPThe Multi-Task GP the latent context embedding multioutput (LCE-M) kernel, with known observation noise.
+DEPRECATED: Please use LCEMGP with train_Yvar instead.
- Parameters:
train_X (Tensor) – (n x d) X training data.
train_Y (Tensor) – (n x 1) Y training data.
-train_Yvar (Tensor) – (n x 1) Noise variances of each training Y.
+train_Yvar (Tensor) – (n x 1) Observed variances of each training Y.
task_feature (int) – Column index of train_X to get context indices.
context_cat_feature (Tensor | None) – (n_contexts x k) one-hot encoded context features. Rows are ordered by context indices, where k is the @@ -2461,21 +2475,21 @@
Models
[burt2020svgp] -(1,2,3,4) +(1,2,3,4)David R. Burt and Carl Edward Rasmussen and Mark van der Wilk, Convergence of Sparse Variational Inference in Gaussian Process Regression, Journal of Machine Learning Research, 2020, http://jmlr.org/papers/v21/19-1015.html.
-[hensman2013svgp] +[hensman2013svgp]James Hensman and Nicolo Fusi and Neil D. Lawrence, Gaussian Processes for Big Data, Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, 2013, https://arxiv.org/abs/1309.6835.
[moss2023ipa] -(1,2,3,4) +(1,2,3,4)Henry B. Moss and Sebastian W. Ober and Victor Picheny, Inducing Point Allocation for Sparse Gaussian Processes in High-Throughput Bayesian Optimization,Proceedings of @@ -2595,12 +2609,12 @@
Models class botorch.models.approximate_gp.SingleTaskVariationalGP(train_X, train_Y=None, likelihood=None, num_outputs=1, learn_inducing_points=True, covar_module=None, mean_module=None, variational_distribution=None, variational_strategy=<class 'gpytorch.variational.variational_strategy.VariationalStrategy'>, inducing_points=None, outcome_transform=None, input_transform=None, inducing_point_allocator=None)[source]¶
Bases:
-ApproximateGPyTorchModelA single-task variational GP model following [hensman2013svgp].
+A single-task variational GP model following [hensman2013svgp].
By default, the inducing points are initialized though the -GreedyVarianceReduction of [burt2020svgp], which is known to be +GreedyVarianceReduction of [burt2020svgp], which is known to be effective for building globally accurate models. However, custom inducing point allocators designed for specific down-stream tasks can also be -provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction +provided (see [moss2023ipa] for details), e.g. GreedyImprovementReduction when the goal is to build a model suitable for standard BO.
A single-task variational GP using relatively strong priors on the Kernel hyperparameters, which work best when covariates are normalized to the unit @@ -2712,7 +2726,7 @@
Models
[Eriksson2021saasbo] -(1,2,3) +(1,2,3)D. Eriksson, M. Jankowiak. High-Dimensional Bayesian Optimization with Sparse Axis-Aligned Subspaces. Proceedings of the Thirty- Seventh Conference on Uncertainty in Artificial Intelligence, 2021.
@@ -2773,7 +2787,7 @@ModelsImplementation of the sparse axis-aligned subspace priors (SAAS) model.
The SAAS model uses sparsity-inducing priors to identify the most important parameters. This model is suitable for high-dimensional BO with potentially -hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.
+hundreds of tunable parameters. See [Eriksson2021saasbo] for more details.SaasPyroModel is not a standard BoTorch model; instead, it is used as an input to SaasFullyBayesianSingleTaskGP. It is used as a default keyword argument, and end users are not likely to need to instantiate or modify a @@ -2902,7 +2916,7 @@
ModelsThis model assumes that the inputs have been normalized to [0, 1]^d and that the output has been standardized to have zero mean and unit variance. You can either normalize and standardize the data before constructing the model or use -an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo] +an input_transform and outcome_transform. The SAAS model [Eriksson2021saasbo] with a Matern-5/2 kernel is used by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it isn’t compatible with fit_gpytorch_model.
@@ -3127,7 +3141,7 @@ModelsA fully Bayesian multi-task GP model with the SAAS prior.
This model assumes that the inputs have been normalized to [0, 1]^d and that the output has been stratified standardized to have zero mean and unit variance for -each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data +each task.The SAAS model [Eriksson2021saasbo] with a Matern-5/2 is used as data kernel by default.
You are expected to use fit_fully_bayesian_model_nuts to fit this model as it isn’t compatible with fit_gpytorch_model.
@@ -3581,14 +3595,14 @@Model Components class botorch.models.kernels.orthogonal_additive_kernel.OrthogonalAdditiveKernel(base_kernel, dim, quad_deg=32, second_order=False, batch_shape=None, dtype=None, device=None, coeff_constraint=Positive())[source]¶
Bases:
-KernelOrthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though +
Orthogonal Additive Kernels (OAKs) were introduced in [Lu2022additive], though only for the case of Gaussian base kernels with a Gaussian input data distribution.
The implementation here generalizes OAKs to arbitrary base kernels by using a Gauss-Legendre quadrature approximation to the required one-dimensional integrals involving the base kernels.
-[Lu2022additive] +[Lu2022additive]@@ -3818,7 +3832,7 @@X. Lu, A. Boukouvalas, and J. Hensman. Additive Gaussian processes revisited. Proceedings of the 39th International Conference on Machine Learning. Jul 2022.
Transforms
-[eriksson2021scalable] +[eriksson2021scalable]D. Eriksson, M. Poloczek. Scalable Constrained Bayesian Optimization. International Conference on Artificial Intelligence and Statistics. PMLR, 2021, http://proceedings.mlr.press/v130/eriksson21a.html
@@ -4257,7 +4271,7 @@Transformsclass botorch.models.transforms.outcome.Bilog(outputs=None)[source]¶
Bases:
OutcomeTransformBilog-transform outcomes.
-The Bilog transform [eriksson2021scalable] is useful for modeling outcome +
The Bilog transform [eriksson2021scalable] is useful for modeling outcome constraints as it magnifies values near zero and flattens extreme values.
Bilog-transform outcomes.
-
@@ -4602,7 +4616,7 @@
-
@@ -5538,7 +5552,7 @@
Utilities
QualityFunctionA function measuring the quality of input points as their expected improvement with respect to a conservative baseline. Expectations -are according to the model from the previous BO step. See [moss2023ipa] +are according to the model from the previous BO step. See [moss2023ipa] for details and justification.
- Parameters: @@ -5555,7 +5569,7 @@
Bases:
-InducingPointAllocatorThe inducing point allocator proposed by [burt2020svgp], that +
The inducing point allocator proposed by [burt2020svgp], that greedily chooses inducing point locations with maximal (conditional) predictive variance.
- Parameters:
-
@@ -5580,11 +5594,11 @@
A pivoted Cholesky initialization method for the inducing points, -originally proposed in [burt2020svgp] with the algorithm itself coming from -[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on +originally proposed in [burt2020svgp] with the algorithm itself coming from +[chen2018dpp]. Code is a PyTorch version from [chen2018dpp], based on https://github.com/laming-chen/fast-map-dpp/blob/master/dpp.py but with a small modification to allow the underlying DPP to be defined through its diversity-quality -decomposition,as discussed by [moss2023ipa]. This method returns a greedy +decomposition,as discussed by [moss2023ipa]. This method returns a greedy approximation of the MAP estimate of the specified DPP, i.e. its returns a set of points that are highly diverse (according to the provided kernel_matrix) and have high quality (according to the provided quality_scores).
diff --git a/v/latest/files/batch_mode_cross_validation.ipynb b/v/latest/files/batch_mode_cross_validation.ipynb index eab9277ed9..383bcc8d3d 100644 --- a/v/latest/files/batch_mode_cross_validation.ipynb +++ b/v/latest/files/batch_mode_cross_validation.ipynb @@ -1,329 +1,330 @@ { - "cells": [ - { - "cell_type": "markdown", - "metadata": { - "collapsed": true - }, - "source": [ - "## Application of batch-mode regression to cross-validation\n", - "\n", - "botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Simple_GP_Regression/Simple_Batch_Mode_GP_Regression.ipynb) from GPyTorch:\n", - "\n", - "$$y = \\sin(2\\pi x) + \\epsilon, ~\\epsilon \\sim \\mathcal N(0, 0.2).$$\n", - "\n", - "Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see [ax.modelbridge.cross_validation](https://github.com/facebook/Ax/blob/main/ax/modelbridge/cross_validation.py). However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "import torch\n", - "import math\n", - "\n", - "device = torch.device(\"cpu\")\n", - "dtype = torch.float\n", - "torch.manual_seed(3);" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Initialize the CV dataset\n", - "\n", - "For our training data, we take 20 regularly spaced points on the interval $[0, 1]$ and generate noisy evaluations with an observed noise variance of 0.2. Remember that botorch requires an explicit output dimension." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "sigma = math.sqrt(0.2)\n", - "train_X = torch.linspace(0, 1, 20, dtype=dtype, device=device).view(-1, 1)\n", - "train_Y_noiseless = torch.sin(train_X * (2 * math.pi))\n", - "train_Y = train_Y_noiseless + sigma * torch.randn_like(train_Y_noiseless)\n", - "train_Yvar = torch.full_like(train_Y, 0.2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The botorch function `gen_loo_cv_folds` takes our observed data `train_X`, `train_Y`, `train_Yvar` as input and returns the LOO CV folds in a `CVFolds` object." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "from botorch.cross_validation import gen_loo_cv_folds\n", - "\n", - "cv_folds = gen_loo_cv_folds(train_X=train_X, train_Y=train_Y, train_Yvar=train_Yvar)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The `cv_folds` object contains the data, stored as tensors of appropriate batch shape, necessary to perform 20 CVs of 19 training points and 1 test point. For example, we can check that the shapes of the training inputs and training targets are `b x n x d = 20 x 19 x 1` and `b x n x o = 20 x 19 x 1` respectively, where `o` is the number of outputs." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": {}, - "outputs": [ + "cells": [ { - "data": { - "text/plain": [ - "(torch.Size([20, 20, 1]), torch.Size([20, 20, 1]))" + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Application of batch-mode regression to cross-validation\n", + "\n", + "botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Exact_GPs/Simple_GP_Regression.ipynb) from GPyTorch:\n", + "\n", + "$$y = \\sin(2\\pi x) + \\epsilon, ~\\epsilon \\sim \\mathcal N(0, 0.2).$$\n", + "\n", + "Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see [ax.modelbridge.cross_validation](https://github.com/facebook/Ax/blob/main/ax/modelbridge/cross_validation.py). However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial." ] - }, - "execution_count": 4, - "metadata": { - "bento_obj_id": "140675275377288" - }, - "output_type": "execute_result" - } - ], - "source": [ - "cv_folds.train_X.shape, cv_folds.train_Y.shape" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": {}, - "outputs": [ + }, { - "data": { - "text/plain": [ - "(torch.Size([20, 1, 1]), torch.Size([20, 1, 1]))" + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "import torch\n", + "import math\n", + "\n", + "device = torch.device(\"cpu\")\n", + "dtype = torch.float\n", + "torch.manual_seed(3);" ] - }, - "execution_count": 5, - "metadata": { - "bento_obj_id": "140676807470216" - }, - "output_type": "execute_result" - } - ], - "source": [ - "cv_folds.test_X.shape, cv_folds.test_Y.shape" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that in a situation where the dataset is large, one may not want to perform LOO; in that case, a similar process can be used to perform $k$-fold CV." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Perform LOOCV\n", - "\n", - "We can use the `batch_cross_validation` function to perform LOOCV using batching (meaning that the `b = 20` sets of training data can be fit as `b = 20` separate GP models with separate hyperparameters in parallel through GPyTorch) and return a CVResult tuple with the batched `GPyTorchPosterior` object over the LOOCV test points and the observed targets. The `batch_cross_validation` requires a model class (`model_cls`) and a marginal log likelihood class (`mll_cls`). Since we have an observed and constant noise level, we will use the FixedNoiseGP as the `model_cls` and an ExactMarginalLogLikelihood as the `mll_cls`." - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "from botorch.cross_validation import batch_cross_validation\n", - "from botorch.models import FixedNoiseGP\n", - "from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood\n", - "\n", - "# instantiate and fit model\n", - "cv_results = batch_cross_validation(\n", - " model_cls=FixedNoiseGP,\n", - " mll_cls=ExactMarginalLogLikelihood,\n", - " cv_folds=cv_folds,\n", - ")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Compute the cross-validation error and generate plots\n", - "To compute the cross-validation error, we first evaluate the test points by computing the posterior in batch mode. Next, we compute the squared errors for each test point from the prediction and take an average across all cross-validation folds." - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": {}, - "outputs": [ + }, { - "name": "stdout", - "output_type": "stream", - "text": [ - "Cross-validation error: 0.062\n" - ] + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Initialize the CV dataset\n", + "\n", + "For our training data, we take 20 regularly spaced points on the interval $[0, 1]$ and generate noisy evaluations with an observed noise variance of 0.2. Remember that botorch requires an explicit output dimension." + ] }, { - "data": { - "image/png": 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RCU6g5HC8M49zxV5DW53/fUBCmGKTGFN+sD1/N/a8vDwGDhxIVlYWSUlJpKen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\niFSi5CAiIpUoOYiISCWeHJU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- "text/plain": [ - "" + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "sigma = math.sqrt(0.2)\n", + "train_X = torch.linspace(0, 1, 20, dtype=dtype, device=device).view(-1, 1)\n", + "train_Y_noiseless = torch.sin(train_X * (2 * math.pi))\n", + "train_Y = train_Y_noiseless + sigma * torch.randn_like(train_Y_noiseless)\n", + "train_Yvar = torch.full_like(train_Y, 0.2)" ] - }, - "metadata": { - "bento_obj_id": "140674656098008" - }, - "output_type": "display_data" - } - ], - "source": [ - "from matplotlib import pyplot as plt\n", - "\n", - "%matplotlib inline\n", - "\n", - "posterior = cv_results.posterior\n", - "mean = posterior.mean\n", - "cv_error = ((cv_folds.test_Y.squeeze() - mean.squeeze()) ** 2).mean()\n", - "print(f\"Cross-validation error: {cv_error : 4.2}\")\n", - "\n", - "# get lower and upper confidence bounds\n", - "lower, upper = posterior.mvn.confidence_region()\n", - "\n", - "# scatterplot of predicted versus test\n", - "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n", - "plt.plot([-1.5, 1.5], [-1.5, 1.5], \"k\", label=\"true objective\", linewidth=2)\n", - "\n", - "axes.set_xlabel(\"Actual\")\n", - "axes.set_ylabel(\"Predicted\")\n", - "\n", - "axes.errorbar(\n", - " x=cv_folds.test_Y.numpy().flatten(),\n", - " y=mean.numpy().flatten(),\n", - " xerr=1.96 * sigma,\n", - " yerr=((upper - lower) / 2).numpy().flatten(),\n", - " fmt=\"*\",\n", - ");" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Finally, we can visualize the fitted models. To do this, we again take advantage of batch-mode evaluation to obtain predictions, including lower and upper confidence regions, from each of the 20 models." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [ - "model = cv_results.model\n", - "with torch.no_grad():\n", - " # evaluate the models at a series of points for plotting\n", - " plot_x = (\n", - " torch.linspace(0, 1, 101).view(1, -1, 1).repeat(cv_folds.train_X.shape[0], 1, 1)\n", - " )\n", - " posterior = model.posterior(plot_x)\n", - " mean = posterior.mean\n", - "\n", - " # get lower and upper confidence bounds\n", - " lower, upper = posterior.mvn.confidence_region()\n", - " plot_x.squeeze_()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The code snippet below plots the result for the 12th CV fold (by setting `num = 12`), but note that we have computed the results for all folds above (other plots can be obtained by iterating `num` from 1 to 20)." - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [ + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The botorch function `gen_loo_cv_folds` takes our observed data `train_X`, `train_Y`, `train_Yvar` as input and returns the LOO CV folds in a `CVFolds` object." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "from botorch.cross_validation import gen_loo_cv_folds\n", + "\n", + "cv_folds = gen_loo_cv_folds(train_X=train_X, train_Y=train_Y, train_Yvar=train_Yvar)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The `cv_folds` object contains the data, stored as tensors of appropriate batch shape, necessary to perform 20 CVs of 19 training points and 1 test point. For example, we can check that the shapes of the training inputs and training targets are `b x n x d = 20 x 19 x 1` and `b x n x o = 20 x 19 x 1` respectively, where `o` is the number of outputs." + ] + }, { - "data": { - "image/png": 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CzJDPBx/tqeRQtTp9iQRlVDTY2VfZpHUY3dLnk7vPB/LfM8jId3Lmj5T/qDU8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\nxVG+OGylxqZOb50am4MPdpb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- "text/plain": [ - " " + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(torch.Size([20, 20, 1]), torch.Size([20, 20, 1]))" + ] + }, + "execution_count": 4, + "metadata": { + "bento_obj_id": "140675275377288" + }, + "output_type": "execute_result" + } + ], + "source": [ + "cv_folds.train_X.shape, cv_folds.train_Y.shape" ] - }, - "metadata": { - "bento_obj_id": "140674628290824" - }, - "output_type": "display_data" + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(torch.Size([20, 1, 1]), torch.Size([20, 1, 1]))" + ] + }, + "execution_count": 5, + "metadata": { + "bento_obj_id": "140676807470216" + }, + "output_type": "execute_result" + } + ], + "source": [ + "cv_folds.test_X.shape, cv_folds.test_Y.shape" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that in a situation where the dataset is large, one may not want to perform LOO; in that case, a similar process can be used to perform $k$-fold CV." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Perform LOOCV\n", + "\n", + "We can use the `batch_cross_validation` function to perform LOOCV using batching (meaning that the `b = 20` sets of training data can be fit as `b = 20` separate GP models with separate hyperparameters in parallel through GPyTorch) and return a CVResult tuple with the batched `GPyTorchPosterior` object over the LOOCV test points and the observed targets. The `batch_cross_validation` requires a model class (`model_cls`) and a marginal log likelihood class (`mll_cls`). Since we have an observed and constant noise level, we will use the FixedNoiseGP as the `model_cls` and an ExactMarginalLogLikelihood as the `mll_cls`." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "from botorch.cross_validation import batch_cross_validation\n", + "from botorch.models import FixedNoiseGP\n", + "from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood\n", + "\n", + "# instantiate and fit model\n", + "cv_results = batch_cross_validation(\n", + " model_cls=FixedNoiseGP,\n", + " mll_cls=ExactMarginalLogLikelihood,\n", + " cv_folds=cv_folds,\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Compute the cross-validation error and generate plots\n", + "To compute the cross-validation error, we first evaluate the test points by computing the posterior in batch mode. Next, we compute the squared errors for each test point from the prediction and take an average across all cross-validation folds." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Cross-validation error: 0.062\n" + ] + }, + { + "data": { + "image/png": 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", + "text/plain": [ + " " + ] + }, + "metadata": { + "bento_obj_id": "140674656098008" + }, + "output_type": "display_data" + } + ], + "source": [ + "from matplotlib import pyplot as plt\n", + "\n", + "%matplotlib inline\n", + "\n", + "posterior = cv_results.posterior\n", + "mean = posterior.mean\n", + "cv_error = ((cv_folds.test_Y.squeeze() - mean.squeeze()) ** 2).mean()\n", + "print(f\"Cross-validation error: {cv_error : 4.2}\")\n", + "\n", + "# get lower and upper confidence bounds\n", + "lower, upper = posterior.mvn.confidence_region()\n", + "\n", + "# scatterplot of predicted versus test\n", + "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n", + "plt.plot([-1.5, 1.5], [-1.5, 1.5], \"k\", label=\"true objective\", linewidth=2)\n", + "\n", + "axes.set_xlabel(\"Actual\")\n", + "axes.set_ylabel(\"Predicted\")\n", + "\n", + "axes.errorbar(\n", + " x=cv_folds.test_Y.numpy().flatten(),\n", + " y=mean.numpy().flatten(),\n", + " xerr=1.96 * sigma,\n", + " yerr=((upper - lower) / 2).numpy().flatten(),\n", + " fmt=\"*\",\n", + ");" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finally, we can visualize the fitted models. To do this, we again take advantage of batch-mode evaluation to obtain predictions, including lower and upper confidence regions, from each of the 20 models." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "model = cv_results.model\n", + "with torch.no_grad():\n", + " # evaluate the models at a series of points for plotting\n", + " plot_x = (\n", + " torch.linspace(0, 1, 101).view(1, -1, 1).repeat(cv_folds.train_X.shape[0], 1, 1)\n", + " )\n", + " posterior = model.posterior(plot_x)\n", + " mean = posterior.mean\n", + "\n", + " # get lower and upper confidence bounds\n", + " lower, upper = posterior.mvn.confidence_region()\n", + " plot_x.squeeze_()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The code snippet below plots the result for the 12th CV fold (by setting `num = 12`), but note that we have computed the results for all folds above (other plots can be obtained by iterating `num` from 1 to 20)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + " " + ] + }, + "metadata": { + "bento_obj_id": "140674628290824" + }, + "output_type": "display_data" + } + ], + "source": [ + "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n", + "\n", + "# plot the 12th CV fold\n", + "num = 12\n", + "\n", + "# plot the training data in black\n", + "axes.plot(\n", + " cv_folds.train_X[num - 1].detach().numpy(),\n", + " cv_folds.train_Y[num - 1].detach().numpy(),\n", + " \"k*\",\n", + ")\n", + "\n", + "# plot the test data in red\n", + "axes.plot(\n", + " cv_folds.test_X[num - 1].detach().numpy(),\n", + " cv_folds.test_Y[num - 1].detach().numpy(),\n", + " \"r*\",\n", + ")\n", + "\n", + "# plot posterior means as blue line\n", + "axes.plot(plot_x[num - 1].numpy(), mean[num - 1].numpy(), \"b\")\n", + "\n", + "# shade between the lower and upper confidence bounds\n", + "axes.fill_between(\n", + " plot_x[num - 1].numpy(), lower[num - 1].numpy(), upper[num - 1].numpy(), alpha=0.5\n", + ");" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "fileHeader": "", + "kernelspec": { + "display_name": "python3", + "language": "python", + "name": "python3" } - ], - "source": [ - "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n", - "\n", - "# plot the 12th CV fold\n", - "num = 12\n", - "\n", - "# plot the training data in black\n", - "axes.plot(\n", - " cv_folds.train_X[num - 1].detach().numpy(),\n", - " cv_folds.train_Y[num - 1].detach().numpy(),\n", - " \"k*\",\n", - ")\n", - "\n", - "# plot the test data in red\n", - "axes.plot(\n", - " cv_folds.test_X[num - 1].detach().numpy(),\n", - " cv_folds.test_Y[num - 1].detach().numpy(),\n", - " \"r*\",\n", - ")\n", - "\n", - "# plot posterior means as blue line\n", - "axes.plot(plot_x[num - 1].numpy(), mean[num - 1].numpy(), \"b\")\n", - "\n", - "# shade between the lower and upper confidence bounds\n", - "axes.fill_between(\n", - " plot_x[num - 1].numpy(), lower[num - 1].numpy(), upper[num - 1].numpy(), alpha=0.5\n", - ");" - ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": true - }, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "python3", - "language": "python", - "name": "python3" - } - }, - "nbformat": 4, - "nbformat_minor": 2 + "nbformat": 4, + "nbformat_minor": 2 } diff --git a/v/latest/files/batch_mode_cross_validation.py b/v/latest/files/batch_mode_cross_validation.py index f87343a3f0..e6ce037214 100644 --- a/v/latest/files/batch_mode_cross_validation.py +++ b/v/latest/files/batch_mode_cross_validation.py @@ -3,7 +3,7 @@ # ## Application of batch-mode regression to cross-validation # -# botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Simple_GP_Regression/Simple_Batch_Mode_GP_Regression.ipynb) from GPyTorch: +# botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Exact_GPs/Simple_GP_Regression.ipynb) from GPyTorch: # # $$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$ # diff --git a/v/latest/js/searchindex.js b/v/latest/js/searchindex.js index 35689eb4be..92893868cb 100644 --- a/v/latest/js/searchindex.js +++ b/v/latest/js/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["acquisition", "cross_validation", "exceptions", "fit", "generation", "index", "logging", "models", "optim", "posteriors", "sampling", "settings", "test_functions", "utils"], "filenames": ["acquisition.rst", "cross_validation.rst", "exceptions.rst", "fit.rst", "generation.rst", "index.rst", "logging.rst", "models.rst", "optim.rst", "posteriors.rst", "sampling.rst", "settings.rst", "test_functions.rst", "utils.rst"], "titles": ["botorch.acquisition", 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"botorch.utils.testing.MockPosterior.variance"]], "zero_grad_ctx() (in module botorch.utils.context_managers)": [[13, "botorch.utils.context_managers.zero_grad_ctx"]]}}) \ No newline at end of file diff --git a/v/latest/tutorials/batch_mode_cross_validation.html b/v/latest/tutorials/batch_mode_cross_validation.html index 4c92ff0202..786f41e9b7 100644 --- a/v/latest/tutorials/batch_mode_cross_validation.html +++ b/v/latest/tutorials/batch_mode_cross_validation.html @@ -72,7 +72,7 @@ diff --git a/v/latest/tutorials/batch_mode_cross_validation/index.html b/v/latest/tutorials/batch_mode_cross_validation/index.html index 4c92ff0202..786f41e9b7 100644 --- a/v/latest/tutorials/batch_mode_cross_validation/index.html +++ b/v/latest/tutorials/batch_mode_cross_validation/index.html @@ -72,7 +72,7 @@@@ -566,313 +355,7 @@-@@ -273,218 +273,7 @@Application of batch-mode regression to cross-validation¶
botorch provides a helper function
+gen_loo_cv_foldsto easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:Application of batch-mode regression to cross-validation¶
botorch provides a helper function
gen_loo_cv_foldsto easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:$$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$
Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see ax.modelbridge.cross_validation. However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial.
Compute the cross
-
+
Compute the cross
-
+
@@ -566,313 +355,7 @@-@@ -273,218 +273,7 @@Application of batch-mode regression to cross-validation¶
botorch provides a helper function
+gen_loo_cv_foldsto easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:Application of batch-mode regression to cross-validation¶
botorch provides a helper function
gen_loo_cv_foldsto easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression tutorial from GPyTorch:$$y = \sin(2\pi x) + \epsilon, ~\epsilon \sim \mathcal N(0, 0.2).$$
Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see ax.modelbridge.cross_validation. However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial.
Compute the cross
-
+
Compute the cross
-
+
Utilities botorch.models.utils.inducing_point_allocators._pivoted_cholesky_init(train_inputs, kernel_matrix, max_length, quality_scores, epsilon=1e-06)[source]¶
Transforms[Daulton2022bopr]. This transformation supports differentiable +proposed in [Daulton2022bopr]. This transformation supports differentiable approximate rounding (currently only for integers). The rounding function is approximated with a piece-wise function where each piece is a hyperbolic tangent function. @@ -4774,8 +4788,8 @@
Transforms[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition -function introduced in [Cakmak2020risk] can be found at +[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition +function introduced in [Cakmak2020risk] can be found at https://botorch.org/tutorials/risk_averse_bo_with_environmental_variables.
The steps for using this to obtain samples of a risk measure are as follows:
-
@@ -5480,7 +5494,7 @@
Utilities
[chen2018dpp] -(1,2) +(1,2)Laming Chen and Guoxin Zhang and Hanning Zhou, Fast greedy MAP inference for determinantal point process to improve recommendation diversity, Proceedings of the 32nd International Conference on Neural Information @@ -5530,7 +5544,7 @@
Utilities[burt2020svgp]. +of [burt2020svgp].
Utilities class botorch.models.utils.inducing_point_allocators.GreedyVarianceReduction[source]¶
Utilities
InducingPointAllocatorAn inducing point allocator that greedily chooses inducing points with large predictive variance and that are in promising regions of the search -space (according to the model form the previous BO step), see [moss2023ipa].
+space (according to the model form the previous BO step), see [moss2023ipa].
Utilities botorch.models.utils.inducing_point_allocators._pivoted_cholesky_init(train_inputs, kernel_matrix, max_length, quality_scores, epsilon=1e-06)[source]¶
Transforms[Daulton2022bopr]. This transformation supports differentiable +proposed in [Daulton2022bopr]. This transformation supports differentiable approximate rounding (currently only for integers). The rounding function is approximated with a piece-wise function where each piece is a hyperbolic tangent function. @@ -4774,8 +4788,8 @@
Transforms[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition -function introduced in [Cakmak2020risk] can be found at +[Cakmak2020risk]. A tutorial notebook implementing the rhoKG acqusition +function introduced in [Cakmak2020risk] can be found at https://botorch.org/tutorials/risk_averse_bo_with_environmental_variables.
The steps for using this to obtain samples of a risk measure are as follows:
-
@@ -5480,7 +5494,7 @@
Utilities
[chen2018dpp] -(1,2) +(1,2)Laming Chen and Guoxin Zhang and Hanning Zhou, Fast greedy MAP inference for determinantal point process to improve recommendation diversity, Proceedings of the 32nd International Conference on Neural Information @@ -5530,7 +5544,7 @@
Utilities[burt2020svgp]. +of [burt2020svgp].
Utilities class botorch.models.utils.inducing_point_allocators.GreedyVarianceReduction[source]¶
Utilities
InducingPointAllocatorAn inducing point allocator that greedily chooses inducing points with large predictive variance and that are in promising regions of the search -space (according to the model form the previous BO step), see [moss2023ipa].
+space (according to the model form the previous BO step), see [moss2023ipa].